The angles of depression of the top and bottom of 8 m tall building from the top of a multistoried building are 30° and 45° respectively. Find the height of the multistoried building and the distance between the two buildings.
Answers
The height of the multistoried building = 4 (√3 + 3) m
The distance between the two buildings = 4 (3+√3 ) m
Consider the figure while going through the following steps:
tan B = PD/BD
tan 30° = PD/BD
1/√3 = PD/BD
BD = PD√3 .............(1)
tan A = PC/AC
tan 45° = PC/BD
1 = PC/BD
BD = PC ............(2)
From (1) and (2), we have
PD√3 = PC
PD√3 = PD + DC
PD√3 - PD = 8
PD (√3 - 1) = 8
PD = 8/(√3 - 1)
upon rationalizing we get,
∴ PD = 4(√3+1)
PC = PD + DC
= 4(√3+1) + 8
= 4√3 + 4 + 8
= 4√3 + 12
∴ PC = 4 (√3 + 3)
The height of the multistoried building = 4 (√3 + 3) m
From (1)
BD = PD√3
= 4(√3+1) (√3 )
= 4 (3+√3 )
∴ AC = BD = 4 (3+√3 )
∴ AC = 4 (3+√3 )
The distance between the two buildings = 4 (3+√3 ) m
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